Recently nearly exact expressions for the distortion in a commonly used family of Pulse Width Modulators (PWMs) known as Asynchronous Sigma Delta Modulators (ASDMs) were presented. Such an ASDM consists of a feedback loop with a schmitt-trigger (or a comparator), and a continuous time loop filter. However these previous results are not yet practically applicable because the effect of unavoidable loop delay (e.g. in the schmitt trigger) was not taken into account. Therefore we now present a more general theory that is also valid when there is a nonzero loop delay. A comparison of the resulting equations with computer simulations demonstrated a very good matching, confirming the validness of the theory. This way, a designer can now easily understand the relationship between the loop filter dynamics and the linearity of an ASDM

Babaie Fishani, Amir

De Bock, Maarten

Rombouts, Pieter

English

Amir Babaie-Fishani

Maarten De Bock

Pieter Rombouts

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Analyzing distortion in ASDMs with loop delay

Ghent University Academic Bibliography

1Analyzing Distortion in ASDMs with Loop DelayAmir Babaie-Fishani, Maarten De Bock and Pieter RomboutsThis document is an author’s draft version submitted for publication to the 2014 IEEE Int. Symp. Circuits and Syst. (ISCAS2014).The actual version was published as [1].REFERENCES[1] A. Babaie-Fishani, M. De Bock, and P. Rombouts, “Analyzing Distortion in ASDMs with Loop Delay,” in Proc. IEEE Int. Symp. Circuits and Syst.(ISCAS), 2014, pp. 77–80.Analyzing Distortion in ASDMs with Loop DelayAmir Babaie-Fishani, Maarten De Bock, and Pieter RomboutsGhent University (UGent), Dept. ELIS, Sint-Pietersnieuwstraat 41, 9000 Ghent, BelgiumEmail: Amir.Babaiefishani@elis.ugent.comAbstract—Recently nearly exact expressions for the distortionin a commonly used family of Pulse Width Modulators (PWMs)known as Asynchronous Sigma Delta Modulators (ASDMs) werepresented. Such an ASDM consists of a feedback loop with aschmitt-trigger (or a comparator), and a continuous time loopfilter. However these previous results are not yet practicallyapplicable because the effect of unavoidable loop delay (e.g. inthe schmitt trigger) was not taken into account. Therefore wenow present a more general theory that is also valid when thereis a nonzero loop delay. A comparison of the resulting equationswith computer simulations demonstrated a very good matching,confirming the validness of the theory. This way, a designer cannow easily understand the relationship between the loop filterdynamics and the linearity of an ASDM.I. INTRODUCTIONRecently, various applications of Pulse Width Modula-tors (PWMs) have been presented: e.g. to linearize Analogto Digital converters [1]–[5] or for class-D Power Ampli-fiers [6], [7]. In these applications, the nonlinearity of thePWM mechanism directly translates into (undesired) overallinput-output nonlinearity. Therefore an accurate understandingof the nonlinearity of the PWM mechanism is of directpractical interest. In this paper we study a very popular formof PWM, known as Asynchronous Sigma Delta Modulation(ASDM), which uses a closed loop with a schmitt-trigger (ora comparator) and a continuous time loop filter. We build forthon our previous work [8] where we have already analyzed thedistortion for the case without loop delay. The theory in [8] wasan improvement to a prior work by Roza [9]. Unfortunatelyin practice, some loop delay (e.g. due to the comparator) isunavoidable. Moreover in some cases, an explicit delay mayeven be added intentionally e.g. to control the PWM carrierfrequency. Therefore, here the theory of [8] is generalized tocover the case of a loop with delay as well.II. PREVIOUS THEORY AND ITS LIMITATIONSAccording to [8], the third harmonic distortion of anADSM, as in Fig.1 (a), with a loop-filter of F(ω) and an inputvoltage of v = A sin(µt) is (all voltages normalized to ±1):d3 =pi26(2A2 − 8−A4 +6A2 − 8A4√1−A2)Re{F (ωc)}|F (3µ)| (1)where ωc is the self oscillation frequency of the PWM forv = 0 and d3 is the amplitude of the third harmonic tone,a3 sin(3µt), over the amplitude of the main tone, d3 = a3A .This theory was obtained based on some assumptions. First,the highest order of si in the numerator of F (s) should beexactly one degree less than the denominator to have a stableoscillation. Second, it assumes that all poles and zeros of theΣ e(t)v(t)Σ e(t)v(t)(a)(b)F(ω)-TDF(ω)p(t)jωep(t)Fig. 1. General representation of an ASDM, (a) without extra delay in theloop, and (b) with loop delay.filter are at much lower frequency than ωc, and finally, thefilter should satisfy the following conditions for ω > ωc :Re{F (ω)} ≈ 1(τr.ω)2and, Im{F (ω)} ≈ 1τi.ω(2)where τr and τi are filter dependent constants and are notnecessarily positive and/or real. Then, given a schmitt triggerthreshold voltage of ±Vth we will have ωc = pi2τi·Vth .The theory in [8] only applies to a certain category ofASDMs where the schmitt trigger is ideal (has no delay). Butin an actual circuit, the schmitt trigger always has a delay,modelled as TD in Fig. 1 (b), and as the oscillation frequencyof the PWM increases, the loop delay becomes more importantin determining the oscillation frequency and the distortion.In order to evaluate d3 for the case with delay, the firstsolution that comes to mind is to merge the delay of the schmitttrigger in the loop-filter as a e−TDjω factor and use (1) forcalculating the third harmonic with the new loop-filter, G(ω) =F (ω).e−TDjω. But it can be easily shown that Re{G(ω)} andIm{G(ω)} will not satisfy the conditions in (2).III. PROPOSED APPROACHHere we will revisit the equations in [8] with similarassumptions while considering a delay of TD for the schmitttrigger. Bare in mind that delay anywhere in the loop can bemodelled this way without affecting the spectral qualities ofthe signals.In Fig. 1 (b), v(t) is a band limited input signal, e(t) isthe filtered error signal at the input of the schmitt-trigger, andp(t) is the output square wave (±1) with a pulse width of αand duration of T . As it will be shown later in this section, inthis type of modulation, not only the duty cycle of the outputsquare wave, αT , but also its carrier frequency are dependent onthe input signal. The self oscillation frequency of this PWM fora zero input is called ωc and ω1 = 2piT is the (instantaneous)-VthA1TDt2T/2Ttte(t)P(t)+1-1-A2α/2-α/2t1VthTD-T/2Fig. 2. Typical waveforms of an schmitt trigger based ASDM with delay.carrier frequency which is a function of the input signal, v.According to Fig. 1 (b) we can write the signal at e(t):e(t) = (v − p(t))⊗ f (3)where f(t) is the time domain impulse response of the filter,F (ω), and ⊗ is the convolution.Assuming that the input signal, v, changes at a muchslower rate than the carrier frequency, it can be consideredapproximately constant during each periods of p(t). Then p(t),which is a square wave, can be described by its Fourier series:p(t) = (2αT− 1) + 4× Re{ ∞∑n=1sin(nω1α2)npiejnω1t}(4)The ( 2αT − 1) term in p(t), which will be referred to as uhereafter, is the baseband component of p(t) which includesthe desired signal and its harmonic distortion components, sowe can say u = ( 2αT − 1) or ω1α = pi(u+ 1) .Fig. 2 shows typical waveforms of an ASDM. As the figureshows, the schmitt-trigger toggles with a delay of TD after e(t)reaches its positive or negative thresholds, Vth or −Vth. So ifwe use t = α2 − TD and substitute (4) in (3), we obtain:−Vth = (v − u)⊗ f − 4∞∑n=1(sin(nω1α2)npi)× (5)(Re {F (nω1)} cos(nω1(α2− TD))−Im {F (nω1)} sin(nω1(α2− TD)))and for t = −α2 − TD we will have:Vth = (v − u)⊗ f − 4∞∑n=1(sin(nω1α2)npi)× (6)(Re {F (nω1)} cos(nω1α2+ nω1TD) +Im {F (nω1)} sin(nω1α2+ nω1TD))By adding the last two equations we will have:0 = (v − u)⊗ f − 2∞∑n=1sin(nω1α)npi× (7)(Im{F (nω1)} sinnω1TD + Re{F (nω1)} cosnω1TD)and subtracting them will give:−piVth2=∞∑n=11− cos(nω1α)n× (8)(Im{F (nω1)} cosnω1TD − Re{F (nω1)} sinnω1TD)If the ASDM has a stable oscillation, then we can deducefrom Fig.2 that 2TD ≤ α + TD ≤ T = ω12pi and from that wecan say 0 ≤ ω1α±ω1TD ≤ 2pi . The terms Re{F (nω1)} andIm{F (nω1)} are obtained from (2). This allows us to solvethe series in (8) using [11]. Then (8) will result in:ω1 = ωc · (1− u2) = pi(τiTD − τ2r ) · (1− u2)2τ2r τiVth + τiT2D − 2τ2r TD(9)This equation accurately gives the instantaneous and selfoscillation frequencies, ω1 and ωc, as functions of the outputbase band signal, u, and the parameters of the modulator.Same as above, we can calculate the series in (7) using (2),(9), [11], and the fact that 0 ≤ ω1α± ω1TD ≤ 2pi . With thatall, we can simplify (7) which will give:(u− v)⊗ f = pi2u6(τrωc)2(1− u2) −uT 2D2τ2r+uTDτi(10)This equation gives an accurate relationship between theinput signal, v, and the output baseband signal, u. In order toavoid the need to solve an implicit equation to obtain u as afunction of v, we can simplify (10) using Perturbation Theory.So at first we make a rough first order estimation from (10).We can see that the terms on the right side of the equationare much smaller than the ones on the left, so we can sayu ≈ v. By applying this estimation to (10) we will obtain thissimplified equation which gives u as an explicit function of v:(u− v)⊗ f ≈ pi2v6(τrωc)2(1− v2) −vT 2D2τ2r+vTDτi(11)Now we will apply a sinusoidal input voltage, v = A sinµt,to (11) to obtain the harmonic distortions of the ASDM. Wecan see that the right hand of this equation is an odd functionof v, so we can expect u to only have odd harmonics of v.Therefore, we can replace u with its Fourier expansion:u =∑nan sin(nµt− θn), n = 1, 3, 5, ... (12)Since the terms on the right hand of (11) are much smallerthan those on the left, for the main tone we can say a1 ≈ A. Inthe right hand of (11), only the first term can produce harmonicdistortion. So if we write the Fourier expansion on both sidesof (11) only for the harmonic tones (n = 3, 5, ...) we will have:∑nan sin(nµt− θn)⊗ f ≈ pi26τ2r ω2c∑nKn(A) sin(nµt) (13)where Kn comes from this Fourier integral:Kn(A) =µpi∫ piµ−piµA sin(µt). sin(nµt)1− (A sin(µt))2 dt (14)and from (13) we can deduce for each harmonic that:θn = ∠[F (nµ)], an · |F (nµ)| = pi26τ2r ω2cKn(A) (15)and by considering (2) for ωc, we can find the ratio of themagnitude of the nth harmonic over the main tone:dn =ana1≈ anA=pi26Kn(A)ARe{F (ωc)}|F (nµ)| (16)The integral in (14) can be solved for any of the harmonics.For n = 3, 5, for example, it gives (for A ≤ 1):K3(A) =2A2 − 8−A3 +6A2 − 8A3√1−A2 (17)K5(A) =2A4 − 24A2 + 32−A5 +10A4 − 40A2 + 16A5√1−A2 (18)By substituting (17) in (16) we can find the 3rd harmonicdistortion of an ASDM with loop delay:d3 =pi26(2A2 − 8−A4 +6A2 − 8A4√1−A2)Re{F (ωc)}|F (3µ)| (19)and the term Re{F (ωc)} can be calculated using (9) and (2):d3 =16(2A2 − 8−A4 +6A2 − 8A4√1−A2)1|F (3µ)| × (20)(2τ2r τiVth + τiT2D − 2τ2r TD)2τ2r (τiTD − τ2r )2In ASDMs without loop delay (assuming that a high orderaggressive loop filter is not chosen) the signal at the input ofthe schmitt trigger is limited to ±Vth, e.g. Fig. 3(a). Afterintroducing delay to the loop, a certain amplitude modulationeffect occurs, Fig.3(b). The (above) envelope of this effectcan be calculated by evaluating e(t = −α2 ), see Fig. (2). Byreplacing (4) in (3) and calculating the series we will have:e(−α2) = uTD(TD2τ2r− 1τi)− TDτi+τ2r Vth − 0.5T 2Dτ2r − τiTD(21)IV. IMPLICATIONS AND RESULTSA. ASDM equivalencyA surprising outcome is that (1) and (19) are the same andthat implies that an ASDM with a schmitt trigger thresholdvoltage of Vth and loop delay of TD is equivalent (regarding itsdistortion) with another ASDM that has an identical loop filterand identical oscillation frequency with a threshold voltage ofVth,eq and zero delay where Vth,eq can be calculated using(20) by solving d3(Vth, TD) = d3(Vth,eq, 0). This value ofequivalent threshold happens to be equal to e(−α2 )|u=0 in (21).B. amplitude modulationIn previous works, the amplitude modulation discussedbefore (21) was believed to be a source of distortion. In [10]a method was presented to cancel out this effect through anactive feedback. The threshold voltage of the schmitt triggerwas dynamically reduced to keep the envelope of the signalat the input of the schmitt trigger constant and equal withthe desired threshold voltage. In this way, the modulator’swaveform and behaviour will be similar to one without delay.This method is only applicable when the schmitt trigger isimplemented using op-amp and resistors. However, as it hasbeen shown in this work, the distortion in an ASDM withdelay is the same as an equivalent one without delay and thereis no need for extra circuitry to compensate it, it is enough tochoose a schmitt trigger with a lower threshold voltage fromthe very beginning.0 1 2 3x 10−7−0.2−0.15−0.1−0.0500.050.10.150.2time (sec)integratederrorsignalate(t)0 1 2 3x 10−7−0.2−0.15−0.1−0.0500.050.10.150.2time (sec)integratederrorsignalate(t)Vth,eqVthconstant envelopeamplitude modulation(a) (b)Fig. 3. typical waveforms of two ASDMs, (a) without any delay in the loop,and (b) with loop delay.C. an ASDM with a comparator and loop delayThe theory presented in this work applies to ASDMsthat use a schmitt trigger (with/without delay) as their non-linear element. But the theory is still valid for the case thatthe threshold voltage of the schmitt trigger is zero and theloop delay is non-zero. In other words, it is possible to use(20) to design an ASDM with a comparator and some delayblocks (like digital buffers) which may be designed easier incomparison with a schmitt trigger.V. SIMULATION RESULTSIn this section, we will describe a few simulations toconfirm the proposed theory. In all cases, for every ASDMwith loop delay, an equivalent ASDM without delay (withsimilar loop filter but higher threshold voltage according tosection IV-A) has been simulated. Then the simulation resultsof these two ASDMs are compared with the proposed theory.A. output spectrum, first order loop filterIn the first set of simulations, we will take a look at theoutput spectrum of two equivalent ASDMs. The schematic ofthe two ASDMs is according to Fig. 1 (a) and (b). Theyboth have a first order passive loop filter, F (s) = 11+τs ,with τ = 5ns. For the ASDM with delay, we have chosenTD = 150ps and Vth = 0.1. Therefore, according to sectionIV-A, the equivalent delay-less ASDM should have a thresholdvoltage of Vth,eq = 0.1266.Fig. 4 (a) and (b) show the output spectrum of these twoASDMs. According to [8] and (9) the average carrier frequencyof an ASDM for a sinusoidal input voltage with an amplitudeof A is ω1 = ωc(1 − 0.5A2). For these cases A = 0.25 waschosen, so f1 = 380.5MHz is expected and according to (20),the predicted third harmonic distortion is d3 = 74.9dB. As thefigure shows, there is good match between the two equivalantASDMs and the proposed theory.B. a second order ASDM with comparator and loop delayIn section IV-C we argued that an ASDM that has acomparator as its non-linear element (instead of a schmitttrigger) and some delay in its loop can be equivalent witha delay-less ASDM with schmitt trigger.In order to verify this argument, for this example wehave chosen a second order loop filter, F (s) = 1+sτ1sτ2·sτ3 , with105 106 107 108 109−160−140−120−100−80−60−40−200Frequency (Hz)Magnitude(dBnormalized)input tone3rd harmonic74.4 dBcarrier @380.2 MHz105 106 107 108 109−160−140−120−100−80−60−40−200Frequency (Hz)Magnitude(dBnormalized)input tone3rd harmonic74.9 dBcarrier @380.9 MHz(a)TD = 150 psVth = 0.1 vτ = 5 psVm = 0.25 vVth = 0.1266τ = 5 psVm = 0.25 v(b)Fig. 4. Output spectrum of two equivalent ASDMs with first order loop filter,(a) with some delay in the loop, and (b) without loop delay.τ1 = τ2 = τ3 = 5ns. The first modulator uses a comparator inits loop (Vth = 0) and has a delay of TD = 600ps. Accordingto section IV-A, this modulator should be equivalent withanother ASDM that uses a schmitt trigger with a thresholdvoltage of Vth,eq = 0.1264 and no delay, TD = 0. Accordingto (9) These values correspond to a self oscillation (zero input)carrier frequency of fc = 390MHzIn Fig. 5(a) these two ASDMs have been compared withthe theory in (20). A sinusoidal voltage with a frequency ofµ = 9.75MHz has been been applied to the input of bothmodulators and their third harmonic distortion d3 has beenevaluated over a range of input amplitudes. As the figureverifies, the two ASDMs are actually equivalent regarding theirdistortion and they both match well with the proposed theoryin (20). A similar set of simulations have been carried outin Fig. 5 (b) where the same ASDMs have been applied asinusoidal signal as their input voltage. This time the amplitudeof the input is constant, A = 0.5, and the frequency is swept.Once again a good match between the two modulators andtheory can be seen.VI. CONCLUSIONWe have derived Eq. (20) to predict the 3rd harmonicdistortion of an ASDM with loop delay. We have also arguedthat an ASDM with loop delay is equivalent with a similarASDM that has no delay, but a higher threshold voltage in itsschmitt trigger. The simulation results matched very well withthe analytical results and verified this equivalency.From a designer’s point of view, the main conclusion isthat the distortion is essentially determined by the loop filterin combination with the self oscillation frequency. The delayin the loop affects the self oscillation frequency, but if thedelay is known, this effect can be counter-acted by tuning theschmitt trigger’s threshold voltage and choosing a lower Vth.0 0.2 0.4 0.6 0.8 1−110−100−90−80−70−60−50−40−30input amplitude0 5 10 15 20−95−90−85−80−75−70−65−60−55−50input tone frequency (MHz)thirdharmonicovermaintone(dB)theory (20)ASDM w/o delayASDM with delaytheory (20)ASDM w/o delayASDM with delaythirdharmonicovermaintone(dB)(a) (b)Fig. 5. Comparing the distortion of two equivalent second order ASDMswith theory.From another perspective, for a given loop filter, better lin-earity can be obtained by increasing the oscillation frequency,fc, which can be realized by decreasing the delay or loweringthe threshold voltage of the schmitt trigger (even as low aszero). The price of this approach is of course higher switchingactivity and power consumption.In practice, all of the relevant requirements can be opti-mized using the explicit equations that have been presented inthis paper.ACKNOWLEDGEMENTThis work has been supported by the Fund for ScientificResearch Flanders (FWO-Vlaanderen) Belgium.REFERENCES[1] L. Hernandez, S. Paton, E. Prefasi, “VCO-based sigma delta modulatorwith PWM precoding,” Electron. Lett., vol. 49, no. 10 pp. 558–559,12th May 2011.[2] G.W. Roberts and M. Ali-Bakhshian, “A Brief Introduction to Time-to-Digital and Digital-to-Time Converters,” IEEE Trans. Circuits Syst.-II,vol. 57, no. 3, pp. 153–157, Mar. 2010.[3] B. De Vuyst and P. Rombouts, “A 5-MHz 11-Bit Self-Oscillating SigmaDelta Modulator With a Delay-Based Phase Shifter in 0.025 mm2,”IEEE J. Solid-State Circuits, vol. 46, no. 8, pp. 1919–1927, Aug. 2011.[4] J. Daniels, W. Dehaene, M. S. J. Steyaert, and A. Wiesbauer, “A/DConversion Using Asynchronous Delta-Sigma Modulation and Time-to-Digital Conversion,” IEEE Trans. Circuits Syst.-I, vol. 57, no. 9,pp. 2404–2410, Sep. 2010.[5] A. Babaie Fishani and P. Rombouts, “Continuous time ∆Σ modulationwith PWM pre-coding and binary gm blocks,” Electron. Lett., vol. 48,no. 19, pp. 1187–1188, Sep. 2012.[6] M. Berkhout, “Class-D audio amplifiers in mobile applications,” IEEEJ. Solid-State Circuits, vol. 46, no. 8, pp. 1919–1927, Aug. 2011.[7] T. Piessens and M.S.J. Steyaert, “Behavioral analysis of self-oscillatingclass D line drivers,” IEEE Trans. Circuits Syst.-I, vol. 52, no. 4, pp.706–714, Apr. 2005.[8] A. Babaie-Fishani, B. Van Keymeulen, and P. Rombouts, “AnalyticalExpressions for the Distortion of Asynchronous Sigma–Delta Modula-tors,” IEEE Trans. Circuits Syst.-II: Express Briefs, vol. 60, no. 8, pp.472–476, 2013.[9] E. Roza, “Analog-to-digital conversion via duty-cycle modulation,”IEEE Trans. Circuits Syst.-II, vol. 44, no. 11, pp. 907–914, Nov. 1997.[10] T. Matic, T.Sandvedek, and M. Herceg, “A Method for the Schmitt-Trigger Propagation-Delay Compensation in Asynchronous Sigma-Delta Modulator,” IEEE Trans. Circuits Syst.-II, vol. 59, no. 7, pp.404–408, Jul. 2012.[11] V. Mangulis, Handbook of series for scientists and engineers. NewYork: Academic, 1965.

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Analyzing distortion in ASDMs with loop delay

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